Extreme values, regular variation, and point processes.

*(English)*Zbl 0633.60001
Applied Probability, Vol. 4, New York etc.: Springer-Verlag. XII, 320 p.; DM 145.00 (1987).

The chapters of this book, after the Preliminaries, are: 1. Domains of attraction and norming constants; 2. Quality of convergence; 3. Point processes; 4. Records and extremal processes; and 5. Multivariate extremes.

The Preliminaries expose essentially Karamata’s theory of regular variation and extensions. Chapters 1 and 2 are good expositions of the subjects with recent results. The chapter on point processes seems too heavy - as the author recognizes by saying that “some skimming may be advisable” - because some results seem not having been used.

Chapter 4 is the kernel of the book with a very good exposition of records and extremal processes, with recent results. The last chapter deals with the essentials of multivariate extremes, with some recent results. Now some minor comments on this good book on the probabilistic aspects of random extremes:

a) the reference structure of the chapters is not uniform: references are scattered through the text, except in the kernel chapter (4) where they are essentially concentrated at the end of the introduction (before 4.1);

b) some references are lacking, e.g.; K. N. Chandler, J. R. Stat. Soc., Ser. B 14, 220-228 (1952; Zbl 0047.383), basic and historical for records (ch. 4); and M. FrĂ©chet, Ann. Soc. Polon. 6, 93-122 (1927), for limiting distributions of univariate extremes (ch. 1);

c) in exercise 0.3.4. \(\alpha\) (t) and \(\beta\) (t) should be interchanged for notational coherence with the use of (Khintchine’s) convergence theorem and the other text of 0.3; d) Gnedenko’s expression “relative stability” should be substituted by the more modern one “multiplicative law of large numbers”; the “additive law of large numbers” is not presented as such, although it appears, partially, on p. 9, and so B. de Finetti, Metron 9, 127-138 (1932; Zbl 0004.12101), is a reference lacking, although contained in the classical Gnedenko, 1943 paper;

e) the notation \(f=(1-F)/F!\) for the hazard function or failure rate is confusing, not only because, in general, \(f=F'\) denotes the probability density but also the usual notation is applied in pp. 42 and 71 for \(n=N'\) (N the standard normal distribution); f) the index could have been improved.

But, let it be stated clearly, there are more of such errata notes, the others being only a little less important. To summarize, after these (almost non-)critical remarks, the book is a good one with a very good exposition of the kernel chapter “Records and extremal processes” which is, in large, the research area of the author. The book is, evidently, to be recommended.

The Preliminaries expose essentially Karamata’s theory of regular variation and extensions. Chapters 1 and 2 are good expositions of the subjects with recent results. The chapter on point processes seems too heavy - as the author recognizes by saying that “some skimming may be advisable” - because some results seem not having been used.

Chapter 4 is the kernel of the book with a very good exposition of records and extremal processes, with recent results. The last chapter deals with the essentials of multivariate extremes, with some recent results. Now some minor comments on this good book on the probabilistic aspects of random extremes:

a) the reference structure of the chapters is not uniform: references are scattered through the text, except in the kernel chapter (4) where they are essentially concentrated at the end of the introduction (before 4.1);

b) some references are lacking, e.g.; K. N. Chandler, J. R. Stat. Soc., Ser. B 14, 220-228 (1952; Zbl 0047.383), basic and historical for records (ch. 4); and M. FrĂ©chet, Ann. Soc. Polon. 6, 93-122 (1927), for limiting distributions of univariate extremes (ch. 1);

c) in exercise 0.3.4. \(\alpha\) (t) and \(\beta\) (t) should be interchanged for notational coherence with the use of (Khintchine’s) convergence theorem and the other text of 0.3; d) Gnedenko’s expression “relative stability” should be substituted by the more modern one “multiplicative law of large numbers”; the “additive law of large numbers” is not presented as such, although it appears, partially, on p. 9, and so B. de Finetti, Metron 9, 127-138 (1932; Zbl 0004.12101), is a reference lacking, although contained in the classical Gnedenko, 1943 paper;

e) the notation \(f=(1-F)/F!\) for the hazard function or failure rate is confusing, not only because, in general, \(f=F'\) denotes the probability density but also the usual notation is applied in pp. 42 and 71 for \(n=N'\) (N the standard normal distribution); f) the index could have been improved.

But, let it be stated clearly, there are more of such errata notes, the others being only a little less important. To summarize, after these (almost non-)critical remarks, the book is a good one with a very good exposition of the kernel chapter “Records and extremal processes” which is, in large, the research area of the author. The book is, evidently, to be recommended.

Reviewer: J.Tiago de Oliveira

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60F99 | Limit theorems in probability theory |

60E99 | Distribution theory |